Table Of ContentAMPL Models for
“Not Linear” Optimization
Using “Linear” Solvers
Robert Fourer
AMPL Optimization Inc.
www.ampl.com — +1 773-336-AMPL
EURO-INFORMS Joint International Meeting
Rome — 1-4 July 2013
Session WB-9, Sponsor Presentations
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
Linear or Not?
“Linear” solvers
Linear and convex quadratic objectives &constraints
Continuous or integer variables (or both)
CPLEX, Gurobi, Xpress, MOSEK, SCIP, CBC, . . .
“Not Linear” problems
Objectives & constraints in any other form
Same continuous or integer variables
Goals
Apply linear solvers to not linear problems
Make this as easy as possible
. . . with help from an algebraic modeling language
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
Intro to AMPL
Algebraic modeling language: symbolic data
set SHIFTS; # shifts
param Nsched; # number of schedules;
set SCHEDS = 1..Nsched; # set of schedules
set SHIFT_LIST {SCHEDS} within SHIFTS;
param rate {SCHEDS} >= 0; # pay rates
param required {SHIFTS} >= 0; # staffing requirements
param least_assign >= 0; # min workers on any schedule used
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
AMPL
Algebraic modeling language: symbolic model
var Work {SCHEDS} >= 0 integer;
var Use {SCHEDS} >= 0 binary;
minimize Total_Cost:
sum {j in SCHEDS} rate[j] * Work[j];
subject to Shift_Needs {i in SHIFTS}:
sum {j in SCHEDS: i in SHIFT_LIST[j]} Work[j] >= required[i];
subject to Least_Use1 {j in SCHEDS}:
least_assign * Use[j] <= Work[j];
subject to Least_Use2 {j in SCHEDS}:
Work[j] <= (max {i in SHIFT_LIST[j]} required[i]) * Use[j];
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
AMPL
Explicit data independent of symbolic model
set SHIFTS := Mon1 Tue1 Wed1 Thu1 Fri1 Sat1
Mon2 Tue2 Wed2 Thu2 Fri2 Sat2
Mon3 Tue3 Wed3 Thu3 Fri3 ;
param Nsched := 126 ;
set SHIFT_LIST[1] := Mon1 Tue1 Wed1 Thu1 Fri1 ;
set SHIFT_LIST[2] := Mon1 Tue1 Wed1 Thu1 Fri2 ;
set SHIFT_LIST[3] := Mon1 Tue1 Wed1 Thu1 Fri3 ;
set SHIFT_LIST[4] := Mon1 Tue1 Wed1 Thu1 Sat1 ;
set SHIFT_LIST[5] := Mon1 Tue1 Wed1 Thu1 Sat2 ; .......
param required := Mon1 100 Mon2 78 Mon3 52
Tue1 100 Tue2 78 Tue3 52
Wed1 100 Wed2 78 Wed3 52
Thu1 100 Thu2 78 Thu3 52
Fri1 100 Fri2 78 Fri3 52
Sat1 100 Sat2 78 ;
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
AMPL
Solver independent of model & data
ampl: model sched1.mod;
ampl: data sched.dat;
ampl: let least_assign := 15;
ampl: option solver cplex;
ampl: solve;
CPLEX 12.5.1.0: optimal integer solution; objective 266
20914 MIP simplex iterations
3085 branch-and-bound node
ampl: option omit_zero_rows 1, display_1col 0;
ampl: display Work;
Work [*] :=
6 28 31 9 66 11 89 9 118 18
18 18 36 7 78 26 91 25 119 7
20 9 37 18 82 18 112 27 122 36
;
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
AMPL
Language independent of solver
ampl: option solver gurobi;
ampl: solve;
Gurobi 5.5.0: optimal solution; objective 266
25713 simplex iterations
2528 branch-and-cut nodes
ampl: display Work;
Work [*] :=
1 20 37 36 89 28 101 12 119 7
2 8 71 7 91 16 109 28 122 8
21 36 87 7 95 8 116 17 124 28
;
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
Intro to “Not Linear” Optimization
How do I linearize this?
. . . + cx y + . . . in my objective
where x, y are variables; c is a positive constant
It depends . . .
What kinds of variables are x and y?
Are you minimizing or maximizing?
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
Intro Example
Case 1: Binary, Minimize
Original formulation
param c > 0;
var x binary;
var y binary;
minimize Obj: ... + c * x * y + ...
Linearization
var z;
minimize Obj: ... + c * z + ...
subject to z0Defn: z >= 0;
subject to zxyDefn: z >= x + y — 1;
. . . z can be continuous
(minimization forces it to 0 or 1)
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
Intro Example
Case 1 (cont’d)
Many other reformulations possible
Best choice depends on problem and solver
This one seems the best overall choice
see tests in Jared Erickson’s dissertation:
[email protected]
Extends to product of two linear terms
Multiply them out
Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers
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IEURO/INFORMS Intl Meeting, Rome —1-4 July 2013 —WB-9 Sponsor Presentations
Description:Robert Fourer, AMPL Models for “Not Linear” Optimization Using Linear Solvers. IEURO/INFORMS Intl Meeting, Rome — 1-4 July 2013 — WB-9 Sponsor Presentations. 1 .. But can minimize product of negative powers . more with conics and quadratics later in talk. 23. Case 4 (cont'd). Intro Example
